The goal of the research is to develop a mathematical model of the mental operations used to solve problems requiring geometrical reasoning. The model's data base will consist of gaze patterns and concurrent written and verbal reports of mathematically-sophisticated subjects as they solve geometrical problems, varying in type and difficulty. The orientations of both eyes and the head, recorded with a unique apparatus, the Maryland Revolving-Field Monitor (MRFM) will be used to determine, accurately and precisely, the location of the subjects' binocular gaze relative to the plane in which the text and figures of a geometrical problem are presented. The MRFM's uniqueness lies in the fact that such gaze patterns can be examined as the subject solves the problem naturally with the head, arms and torso free to move while the solution is worked out, narrated and written down. The MRFM requires attachments to the eyes, so its use is limited to a small set of well-informed, adult subjects. The gaze patterns of four, mathematically-sophisticated subjects, who routinely run in eye movement experiments using the MRFM, will provide the data base for developing the math model of interest. Once developed, the applicability of this model to the performance of mathematically- gifted children will be tested with children under study at Stanford. The data base for these children will be their verbal and written reports. The long range goal for this project is to develop a mathematical model of geometrical reasoning, and then use this model to study the development of mathematical sophistication. Attaining this goal has potential practical applications in both education and mental health. Namely, it can be used for (1) diagnosis of intellectual disabilities in the young and following injury to the central nervous system; (2) screening for exceptional mathematical gifts and (3) teaching less gifted individuals more efficient problem solving skills.